Table of contents

This page was last modified on: 06 July 2024, 11:55, CST

Locally flat chains in Euclidean space

U. Menne, C. Scharrer
[15] A priori bounds for geodesic diameter. Part I. Integral chains with coefficients in a complete normed commutative group, 41 pages.
Rev. Mat. Iberoam., accepted, Apr 2024.
ArXiv: 2206.14046v2 [math.DG].
Abstract: As service to the community, we provide—for Euclidean space—a basic treatment of locally rectifiable chains and of the complex of locally integral chains. In this setting, we may beneficially develop the idea of a complete normed commutative group bundle over the Grassmann manifold whose fibre is the coefficient group of the chains. Our exposition also sheds new light on some algebraic aspects of the theory. Finally, we indicate an extension to a geometric approach to locally flat chains centring on locally rectifiable chains rather than completion procedures.

Differentiability theory for distributions and subsets of Euclidean space

U. Menne
[14] Pointwise differentiability of higher-order for distributions, 32 pages.
Anal. PDE, 14(2):323–354, Mar 2021.
DOI: 10.2140/apde.2021.14.323. ArXiv: 1803.10855v2 [math.FA].
Abstract: For distributions, we build a theory of higher order pointwise differentiability comprising, for order zero, Łojasiewicz's notion of point value. Results include Borel regularity of differentials, higher order rectifiability of the associated jets, a Rademacher-Stepanov type differentiability theorem, and a Lusin type approximation. A substantial part of this development is new also for zeroth order. Moreover, we establish a Poincaré inequality involving the natural norms of negative order of differentiability. As a corollary, we characterise pointwise differentiability in terms of point values of distributional partial derivatives.

U. Menne, M. Santilli
[11] A geometric second-order-rectifiable stratification for closed subsets of Euclidean space, 14 pages.
Ann. Sc. Norm. Super. Pisa Cl. Sci. (5), 19(3):1185–1198, Sep 2019.
DOI: 10.2422/2036-2145.201703_021. ArXiv: 1703.09561v2 [math.CA].
Abstract: Defining the m-th stratum of a closed subset of an n dimensional Euclidean space to consist of those points, where it can be touched by a ball from at least n−m linearly independent directions, we establish that the m-th stratum is second-order rectifiable of dimension m and a Borel set. This was known for convex sets, but is new even for sets of positive reach. The result is based on a sufficient condition of parametric type for second-order rectifiability.

U. Menne
[9] Pointwise differentiability of higher order for sets, 31 pages.
Ann. Global Anal. Geom., 55(3), 591–621, Apr 2019.
Springer Nature SharedIt:
DOI: 10.1007/s10455-018-9642-0. ArXiv: 1603.08587v2 [math.DG].
Abstract: The present paper develops two concepts of pointwise differentiability of higher order for arbitrary subsets of Euclidean space defined by comparing their distance functions to those of smooth submanifolds. Results include that differentials are Borel functions, higher order rectifiability of the set of differentiability points, and a Rademacher result. One concept is characterised by a limit procedure involving inhomogeneously dilated sets. The original motivation to formulate the concepts stems from studying the support of stationary integral varifolds. In particular, strong pointwise differentiability of every positive integer order is shown at almost all points of the intersection of the support with a given plane.

Survey on varifolds suitable for graduate students

U. Menne
[12] The concept of varifold, 5 pages.
Notices Amer. Math. Soc., 64(10):1148–1152, Nov 2017.
DOI: 10.1090/noti1589. ArXiv: 1705.05253v4 [math.DG].
Abstract: We survey – by means of 20 examples – the concept of varifold, as generalised submanifold, with emphasis on regularity of integral varifolds with mean curvature, while keeping prerequisites to a minimum. Integral varifolds are the natural language for studying the variational theory of the area integrand if one considers, for instance, existence or regularity of stationary (or, stable) surfaces of dimension at least three, or the limiting behaviour of sequences of smooth submanifolds under area and mean curvature bounds.

Partial differential equations on singular surfaces with mean curvature – foundations

U. Menne
[17] A sharp lower bound on the mean curvature integral with critical power for integral varifolds, 144 pages.
Manuscript dated 2014. Complete proofs, few reader guidance, 24 chapters, 1 appendix, 144 pages. Part I (Chapters 1-14) is published in [6]; Chapter 15 is published as part of [8]; some material of Chapter 19 is included in [9]; other chapters shall be generalised (with collaborators) and published elsewhere.
ArXiv: 2310.01754v1 [math.DG], Oct 2023.
Part I Weakly differentiable functions
Part II PDEs on varifolds
15 Sobolev spaces … 75
16 Preliminaries … 82
17 Maximum estimates … 86
18 Second order flatness of Lebesgue spaces for integral varifolds with subcritical integrability of the mean curvature … 94
19 Second order differentiability of the support of integral varifolds with critical integrability of the mean curvature … 100
20 Harnack inequality … 108
Part III The generalised Gauss map
21 Points of finite lower density … 118
22 Points of infinite density … 130
23 Area formula for the generalised Gauss map … 137
24 A sharp lower bound on the mean curvature integral with critical power for integral varifolds … 138
A Monotonicity identity … 139

U. Menne
[8] Sobolev functions on varifolds, 50 pages.
Proc. Lond. Math. Soc. (3), 113(6):725–774, Dec 2016.
Wiley Content Sharing:
DOI: 10.1112/plms/pdw023. ArXiv: 1509.01178v3 [math.CA].
Abstract: This paper introduces first-order Sobolev spaces on certain rectifiable varifolds. These complete locally convex spaces are contained in the generally non-linear class of generalised weakly differentiable functions and share key functional analytic properties with their Euclidean counterparts.
Assuming the varifold to satisfy a uniform lower density bound and a dimensionally critical summability condition on its mean curvature, the following statements hold. Firstly, continuous and compact embeddings of Sobolev spaces into Lebesgue spaces and spaces of continuous functions are available. Secondly, the geodesic distance associated to the varifold is a continuous, not necessarily Hölder continuous Sobolev function with bounded derivative. Thirdly, if the varifold additionally has bounded mean curvature and finite measure, then the present Sobolev spaces are isomorphic to those previously available for finite Radon measures yielding many new results for those classes as well.
Suitable versions of the embedding results obtained for Sobolev functions hold in the larger class of generalised weakly differentiable functions.

U. Menne
[6] Weakly differentiable functions on varifolds, 112 pages.
Indiana Univ. Math. J., 65(3):977–1088, Jul 2016.
DOI: 10.1512/iumj.2016.65.5829. ArXiv: 1411.3287v1 [math.DG].
Abstract: The present paper is intended to provide the basis for the study of weakly differentiable functions on rectifiable varifolds with locally bounded first variation. The concept proposed here is defined by means of integration-by-parts identities for certain compositions with smooth functions. In this class, the idea of zero boundary values is realised using the relative perimeter of superlevel sets. Results include a variety of Sobolev Poincaré-type embeddings, embeddings into spaces of continuous and sometimes Hölder-continuous functions, and pointwise differentiability results both of approximate and integral type as well as coarea formulae.
As a prerequisite for this study, decomposition properties of such varifolds and a relative isoperimetric inequality are established. Both involve a concept of distributional boundary of a set introduced for this purpose.
As applications, the finiteness of the geodesic distance associated with varifolds with suitable summability of the mean curvature and a characterisation of curvature varifolds are obtained.

Regularity of singular surfaces with mean curvature

S. Kolasiński, U. Menne
[7] Decay rates for the quadratic and super-quadratic tilt-excess of integral varifolds, 56 pages.
NoDEA Nonlinear Differential Equations Appl., 24:Art. 17, 56, Mar 2017.
DOI: 10.1007/s00030-017-0436-z. ArXiv: 1501.07037v2 [math.DG].
Abstract: This paper concerns integral varifolds of arbitrary dimension in an open subset of Euclidean space satisfying integrability conditions on their first variation. Firstly, the study of pointwise power decay rates almost everywhere of the quadratic tilt-excess is completed by establishing the precise decay rate for two-dimensional integral varifolds of locally bounded first variation. In order to obtain the exact decay rate, a coercive estimate involving a height-excess quantity measured in Orlicz spaces is established. Moreover, counter-examples to pointwise power decay rates almost everywhere of the super-quadratic tilt-excess are obtained. These examples are optimal in terms of the dimension of the varifold and the exponent of the integrability condition in most cases, for example if the varifold is not two-dimensional. These examples also demonstrate that within the scale of Lebesgue spaces no local higher integrability of the second fundamental form, of an at least two-dimensional curvature varifold, may be deduced from boundedness of its generalised mean curvature vector. Amongst the tools are Cartesian products of curvature varifolds.

U. Menne
[4] Second order rectifiability of integral varifolds of locally bounded first variation, 55 pages.
J. Geom. Anal., 23(2):709–763, Apr 2013.
DOI: 10.1007/s12220-011-9261-5. ArXiv: 0808.3665v3 [math.DG].
Abstract: It is shown that every integral varifold in an open subset of Euclidean space whose first variation with respect to area is representable by integration can be covered by a countable collection of submanifolds of the same dimension of class 2 and that their mean curvature agrees almost everywhere with the variationally defined generalized mean curvature of the varifold.

U. Menne
[3] Decay estimates for the quadratic tilt-excess of integral varifolds, 83 pages.
Arch. Ration. Mech. Anal., 204(1):1–83, Apr 2012.
DOI: 10.1007/s00205-011-0468-1. ArXiv: 0909.3253v3 [math.DG].
Abstract: This paper concerns integral varifolds of arbitrary dimension in an open subset of Euclidean space with their first variation given by either a Radon measure or a function in some Lebesgue space. Pointwise decay results for the quadratic tilt-excess are established for those varifolds. The results are optimal in terms of the dimension of the varifold and the exponent of the Lebesgue space in most cases, for example if the varifold is not two-dimensional.

Basic geometric properties of singular surfaces with mean curvature

U. Menne, C. Scharrer
[16] A priori bounds for geodesic diameter. Part II. Fine connectedness properties of varifolds, 58 pages.
ArXiv: 2209.05955v2 [math.DG], Aug 2023.
Abstract: For varifolds whose first variation is representable by integration, we introduce the notion of indecomposability with respect to locally Lipschitzian real valued functions. Unlike indecomposability, this weaker connectedness property is inherited by varifolds associated with solutions to geometric variational problems phrased in terms of sets, G chains, and immersions; yet it is strong enough for the subsequent deduction of substantial geometric consequences therefrom. Our present study is based on several further concepts for varifolds put forward in this paper: real valued functions of generalised bounded variation thereon, partitions thereof in general, partition thereof along a real valued generalised weakly differentiable function in particular, and local finiteness of decompositions.

U. Menne, C. Scharrer
[13] A priori bounds for geodesic diameter. Part III. A Sobolev-Poincaré inequality and applications to a variety of geometric variational problems, 46 pages
ArXiv: 1709.05504v2 [math.DG], Mar 2024.
Abstract: Based on a novel type of Sobolev-Poincaré inequality (for generalised weakly differentiable functions on varifolds), we establish a finite upper bound of the geodesic diameter of generalised compact connected surfaces-with-boundary of arbitrary dimension in Euclidean space in terms of the mean curvatures of the surface and its boundary. Our varifold setting includes smooth immersions, surfaces with finite Willmore energy, two-convex hypersurfaces in level-set mean curvature flow, integral currents with prescribed mean curvature vector, area minimising integral chains with coefficients in a complete normed commutative group, varifold solutions to Plateau’s problem furnished by min-max methods or by Brakke flow, and compact sets solving Plateau problems based on Čech homology. Due to the generally inevitable presence of singularities, path-connectedness was previously known neither for the class of varifolds (even in the absence of boundary) nor for the solutions to the Plateau problems considered.

U. Menne, C. Scharrer
[10] An isoperimetric inequality for diffused surfaces, 16 pages.
Kodai Math. J., 41(1):70–85, Mar 2018.
DOI: 10.2996/kmj/1521424824. ArXiv: 1612.03823v2 [math.DG].
Abstract: For general varifolds in Euclidean space, we prove an isoperimetric inequality, adapt the basic theory of generalised weakly differentiable functions, and obtain several Sobolev type inequalities. We thereby intend to facilitate the use of varifold theory in the study of diffused surfaces.

U. Menne
[2] A Sobolev Poincaré type inequality for integral varifolds, 40 pages.
Calc. Var. Partial Differential Equations, 38(3-4):369–408, Jul 2010.
DOI: 10.1007/s00526-009-0291-9. ArXiv: 0808.3660v2 [math.DG].
Abstract: In this work a local inequality is provided which bounds the distance of an integral varifold from a multivalued plane (height) by its tilt and mean curvature. The bounds obtained for the exponents of the Lebesgue spaces involved are shown to be sharp.

U. Menne
[1] Some applications of the isoperimetric inequality for integral varifolds, 23 pages.
Adv. Calc. Var., 2(3):247–269, Jul 2009.
DOI: 10.1515/ACV.2009.010. ArXiv: 0808.3652v1 [math.DG].
Abstract: In this work the isoperimetric inequality for integral varifolds of locally bounded first variation is used to obtain sharp estimates for the size of the set where the density quotient is small and to generalise Calderón's and Zygmund's theory of first order differentiability for functions in Lebesgue spaces from Lebesgue measure to integral varifolds.

Conference proceedings

U. Menne
[5] A sharp lower bound on the mean curvature integral with critical power for integral varifolds, 3 pages.
In abstracts from the workshop held July 22-28, 2012, Organized by Camillo De Lellis, Gerhard Huisken and Robert Jerrard, Oberwolfach Reports. Vol. 9, no. 3, 2012.
DOI: 10.4171/OWR/2012/36.

List of minor corrections

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